Fast Fractal Technique using Modified Moment Features on Domain Blocks

In this research, a new technique is suggested to reduce the long time required by the encoding process by using modified moment features on domain blocks. The modified moment features were used in accelerating the matching step of the Iterated Function System (IFS). The main disadvantage facing the fractal image compression (FIC) method is the over-long encoding time needed for checking all domain blocks and choosing the least error to get the best matched domain for each block of ranges. In this paper, we develop a method that can reduce the encoding time of FIC by reducing the size of the domain pool based on the moment features of domain blocks, followed by a comparison with threshold (the selected threshold based on experience is 0.0001). The experiment was conducted on three images with size of 512x512 pixel, resolution of 8 bits/pixel, and different block size (4x4, 8x8 and, 16x16 pixels). The resulted encoding time (ET) values achieved by the proposed method were 41.53, 39.06, and 38.16 sec, respectively, for boat , butterfly, and house images of block size 4x4 pixel. These values were compared with those obtained by the traditional algorithm for the same images with the same block size, which were 1073.85, 1102.66, and 1084.92 sec, respectively. The results imply that the proposed algorithm could remarkably reduce the ET of the images in comparison with the traditional algorithm.

On the group of isometries of planar IFS fractals*

For any iterated function system (IFS) on R 2 , let K be the attractor. Consider the group of all isometries on K. If K is a self-similar or self-affine set, it is proven that the group must be finite. If K is a bi-Lipschitz IFS fractal, the necessary and sufficient conditions for the infiniteness (or finiteness) of the group are given. For the finite case, the computation of the size of the group is also discussed.

Morphogenesis Analysis for Digital Image Production with L-System

The process of forming an image requires a correct color composition, location and distance between the lines to produce a good image. Human abilities in both creativity and high imagination are very limited, especially in forming new images by utilizing existing image patterns or images that resemble old images. Here we showed the implementation of L-System to generate new image generations with additional flame as a fire effect/glow on images for image transformation. This research used the L-System algorithm, Iterated Function System, and Voronoi Diagram to improve the result of image transformation. The results of this study indicated that mathematical calculations can be applied in the formation of images and the resulting images can be abstract and symmetrical. The next generation of images produced in this research can be in unlimited numbers as the generation of morphogenesis processes. The process of generating images is carried out randomly by merging the two existing images with morphogenesis analogy. The resulting images can be exported into jpg, png, and svg formats. Furthermore, this research showed that the implementation of the calculation for the variation reach the value of 99.48% while the image variation composition has a value of 99.29%.

Some properties of the fractal convolution of functions

We consider the fractal convolution of two maps f and g defined on a real interval as a way of generating a new function by means of a suitable iterated function system linked to a partition of the interval. Based on this binary operation, we consider the left and right partial convolutions, and study their properties. Though the operation is not commutative, the one-sided convolutions have similar (but not equal) characteristics. The operators defined by the lateral convolutions are both nonlinear, bi-Lipschitz and homeomorphic. Along with their self-compositions, they are Fréchet differentiable. They are also quasi-isometries under certain conditions of the scale factors of the iterated function system. We also prove some topological properties of the convolution of two sets of functions. In the last part of the paper, we study stability conditions of the dynamical systems associated with the one-sided convolution operators.

Generalized modular fractal spaces and fixed point theorems

In this article, we introduce a new concept of Hausdorff distance through generalized modular metric on nonempty compact subsets and study some topological properties of it. This concept with contraction theory and the iterated function system (IFS) helps us to define a generalized modular fractal space.

A Novel Iterated Function System-Based Model for Coloured Image Encryption

Chaotic cryptography has been a well-studied domain over the last few years. Many works have been done, and the researchers are still getting benefit from this incredible mathematical concept. This paper proposes a new model for coloured image encryption using simple but efficient chaotic equations. The proposed model consists of a symmetric encryption scheme in which it uses the logistic equation to generate secrete keys then an affine recursive transformation to encrypt pixels' values. The experimentations show good results, and theoretic discussion proves the efficiency of the proposed model.